Alisson has played a key role in Liverpool's Premier League and Champions League triumphs since signing from Roma in 2018.
The 34-year-old Brazilian has made 347 appearances for the Reds, winning seven major trophies.
Alisson is out of contract at the end of the current campaign and has been linked with a move, but is keen to prolong his time on Merseyside with a new deal.
"Yes (I want to stay), in my mind. I'm focused on playing every game, but in my mind I'm happy to stay," Alisson said after Sunday's 3-1 win at Crystal Palace in the Premier League.
"That depends on a lot of aspects and different situations. But my will is to stay. I'm happy here and my family is happy here.
"I'm still hungry for titles and to (keep) delivering good performances. I still want to play at the top level for as long as I can, so that's my thinking."
Liverpool boss Andoni Iraola recently said he hoped the 'keeper could "continue for more time" with his team.
Alisson revealed "my agent is talking with the club" after a close season that featured talk of a potential switch to Juventus.
He underlined his enduring value with a fine performance to help Liverpool beat Palace.
But Giorgi Mamardashvili is waiting in the wings as Alisson's solid deputy, with teenage 'keeper Lucca Brughmans set to join Liverpool from Genk next year.
"Our reality is that we're not doing as well as we can. The consistency is not there yet," Alisson said of Liverpool's inconsistent start to the season.
"We have a top manager, top players. We are capable of great things, but as a team we need to get to a point where we feel: 'We are OK now. We have the confidence. We have everything that's needed to win.'"